Quasilinear stabilization of the free electron laser instability for a relativistic electron beam propagating through a transverse helical wiggler magnetic field

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Abstract
A quasilinear model is developed that describes the nonlinear evolution and stabilization of the free electron laser instability in circumstances where a broad spectrum of waves is excited. The relativistic electron beam propagates perpendicular to a helical wiggler magnetic field B0=−B̂ cos k0 z êx−B̂ sin k0 z êy, and the analysis is based on the Vlasov–Maxwell equations assuming ∂/∂x=0=∂/∂ y and a sufficiently tenuous beam that the Compton‐regime approximation is valid (δφ≂0). Coupled kinetic equations are derived that describe the evolution of the average distribution function G0( pz,t) and spectral energy density Ek(t) in the amplifying electromagnetic field perturbations. A thorough exposition of the theoretical model and general quasilinear formalism is presented, and the stabilization process is examined in detail for weak resonant instability with small temporal growth rate γk satisfying ‖γk/ωk‖≪1 and ‖γk/k Δvz‖≪1. Assuming that the beam electrons have small fractional momentum spread (Δ pz/p0≪1), the process of quasilinear stabilization by plateau formation in the resonant region of velocity space (ωk−kvz=0) is investigated, including estimates of the saturated field energy, efficiency of radiation generation, etc.